A 'plane' is a perfectly flat surface defined mathematically by three points. A planer is designed to create a perfectly flat surface on wood.
If you mean a planer and a shaper, as in the woodworking tools, there are many differences to discuss.The most obvious is that a planer is normally handheld and shaves small layers of wood off of timber for squaring and other simple purposes. A shaper, on the other hand, is a large machine that is used for carving panels in doors and cabinetry.A planer is generally safer and simpler to operate. I'm no expert in woodworking--you'd be better off going to another site or asking someone who teaches Woodworking.
A thickness planer is a type of woodworking machine. This machine is used to make boards that are of the same thickness throughout the length of the board as well as being flat on both of the surfaces.
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There is the chisel family, screwdriver family (straight, Phillips, etc.); planer family, sandpaper (different grades), saws (crosscut, box, coping, for a few) ...and more tools.
They're called wood "Rasps".
Combination machines are generally woodworking machines. There can be different variants of the combination. One machine would be a combination of a table saw with a jointer for example. Another would be a combination of a jointer and a planer.
Points that are within the same plane are called co-planer.
Maddie Planer's birth name is Madeline Barnes Planer.
Nigel Planer's birth name is Nigel George Planer.
Maddie Planer is 5' 3".
Nigel Planer is 6' 3".
A board towed behind a boat is called a Planer.